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A piece of paper cut into the shape shown can be folded into a cube with a side length \(l\).

  1. Assuming that the shape can be cut from the piece of paper, what is the minimum area of a piece of paper that can be used to make a cube with side length \(l\)?
    The surface area of the shape is \(6l^2\), so the least area will be \(6l^2\).

  2. Graphically represent all areas of paper that could be used to make a cube with side length \(l\).

    The minimum area is \(6l^2\), so \(A \ge 6l^2\).



  3. Select a point from the solution region and explain why it is a solution.

    The point \((1, 7)\) is a solution. In this case, the length is \(1\) which means the area is \(6l^2 = 6(1)^2 = 6\). There is an area of \(7\), so there is enough paper to make this cube.

 For more examples on quadratic inequalities in two variables, see pp. 488 – 496 of Pre-Calculus 11.